We can understand this behavior by modeling the QPC as a source of electrical, high-frequency noise coupling to the DQD detuning

δ. Consider a DQD capacitively coupled to an alternating voltage source shaking the DQD detuning according to

$e\widehat{V}\left(t\right)$. The tunneling rate between the two dots will generally be influenced by the voltage, an effect which is termed photon-assisted tunneling (PAT) [

23] and has been studied e.g., in Reference [

24], where the voltage is a discrete-frequency microwave signal applied to one of the gates of the sample.

In this case, as well as in our experiment, the energy transfer from the source (the microwave in the coaxial cable, or the QPC, respectively) and the detector (DQD) takes place in the near field,

i.e., the distance between the two is small compared to the radiation wavelength. No free propagation of photons occurs, and it is therefore possible to describe the process of photon-assisted tunneling in terms of a capacitive (in essence electrostatic) coupling. Nevertheless, the term “photon” is commonly used in this context and hints to the fact that the scale of the transferred energy is linked to the frequency, not the amplitude

$\widehat{V}$, of the voltage. Namely, if the characteristic frequency of the voltage is much larger than the interdot tunneling rate, we can express the probability to absorb an energy quantum

E from the field

$\widehat{V}\left(t\right)$ during the tunneling process as [

25],

Here, the voltage fluctuations are described in terms of the autocorrelation function

$J\left(t\right)=\langle [\widehat{\varphi}\left(t\right)-\widehat{\varphi}\left(0\right)]\widehat{\varphi}\left(0\right)\rangle $ of the phase operators

$\widehat{\varphi}\left(t\right)={\int}_{0}^{t}\mathrm{d}{t}^{\prime}e\widehat{V}\left({t}^{\prime}\right)/\hslash $. From the leading-order expansion in

$\widehat{V}$ [

26], it follows that

$P\left(E\right)$ is related to the spectral density

${S}_{V}\left(\omega \right)$ of the voltage fluctuations

$\widehat{V}$,

The probability

P can be interpreted as an effective (tunneling) density of the dot states, which, in the absence of the perturbation

${S}_{V}$, is a peaked at zero energy and zero otherwise, allowing only for elastic tunneling processes. At nonzero

${S}_{V}$, also the probability to absorb energy

$E>0$ from the voltage field becomes nonzero. In addition, the weight of the

δ-peak at zero energy is reduced, as the total weight of the dot states is conserved. Note that the spectrum

${S}_{V}\left(\omega \right)$ appearing in Equation

2 is a non-symmetrized one [

27]. The values of

${S}_{V}$ at negative

ω lead to stimulated and spontaneous emission of energy into the voltage field.

In our experiment, the role of the voltage source is played by the QPC. Namely, a QPC biased at finite source-drain voltage generates current shot-noise, or excess noise. Its spectrum

${S}_{I}^{\mathrm{ex}}(\omega ;{V}_{\mathrm{qpc}})$ [

28,

29] has a characteristic cut-off frequency of

$e{V}_{\mathrm{qpc}}/\hslash $, since the photons that are emitted by the QPC in the form of noise have an energy bounded by the bias voltage

$|e{V}_{\mathrm{qpc}}|$. In Reference [

26], a mechanism based on a circuit model is proposed, coupling the current fluctuations in the QPC to voltage fluctuations in

δ via a trans-impedance,

${S}_{V}(\omega ;{V}_{\mathrm{qpc}})={\left|{Z}_{\mathrm{tr}}\right|}^{2}{S}_{I}^{\mathrm{ex}}(\omega ;{V}_{\mathrm{qpc}})$. Using these three ingredients, that is PAT theory, QPC shot-noise theory, and the QPC-DQD coupling mechanism, we can understand the features of the data in

Figure 3(b) discussed above.

In the right panel of

Figure 3(b), we present a quantitative comparison of our data with theory showing good agreement. A coupling impedance of

${Z}_{\mathrm{tr}}=5.4\phantom{\rule{0.166667em}{0ex}}\mathrm{k}\Omega $ was determined as a fitting parameter. In the calculation, which is detailed in Reference [

19], it is also taken into account that there are environmental (equilibrium) fluctuations other than the QPC excess noise, such as phonons in the InAs nanowire. These lead to the thermal broadening of the

${\Gamma}_{\mathrm{interdot}}$ peak. Although not fully under experimental control, the measurement of

${\Gamma}_{\mathrm{interdot}}$ close to zero QPC bias serves as an indirect measurement of the equilibrium fluctuations, and these data enter the calculation shown in

Figure 3(b). The vertical stripes appearing in the colorplot are an artifact of this mixing of experimental and theoretical data: small statistical fluctuations in the measurement of

${\Gamma}_{\mathrm{interdot}}$ are incorporated into the fit and remain visible until they are eventually averaged out at

$|{V}_{\mathrm{qpc}}|\gtrsim 1\phantom{\rule{0.166667em}{0ex}}\mathrm{mV}$.